Whether the given series is convergent or divergent.

Question

It is given that:

$$\sum_{n=1}^\infty a_n=\sum_{n=1}^\infty \frac{(n+1)^n}{n^{\frac{3}{2}+n}}$$

My Attempt: Since there is a power of $n$ involved, I tried root test. It turns out that lim $n$ tending to infinity $|a_{n}|^{1/n}=1$. So, the test is inconclusive.

Then I went for ratio test, but that didn't help me much. I am unable to determine whether the series is convergent or divergent.

Any help is appreciated.Thankyou.

Answer 1

We have

$$\frac{(n+1)^n}{n^{\frac{3}{2}+n}}=\frac{\left(1+\frac1n\right)^n}{n^{\frac{3}{2}}}\to 0$$

Then the series $\sum_{n=1}^\infty \frac{(n+1)^n}{n^{\frac{3}{2}+n}}$ converges by limit comparison test with $\sum_{n=1}^\infty \frac1{n^{\frac{3}{2}}}$ since

$$\frac{a_n}{\frac1{n^{\frac{3}{2}}}}=\left(1+\frac1n\right)^n\to e$$

Answer 2

$a_n=(1+\frac 1 n)^{n}\frac 1 {n^{3/2}} \to (e)(0)=0$.

Answer 3

$\frac{(n+1)^n}{n^n} \frac{1}{n^{\frac{3}{2}}} \to e \times 0=0$